Algorithmic worlds

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2011-04-29 Barbara Takenaga

2011-04-13 Play with Ducks

2011-03-31 Karl Blossfeldt

2011-03-30 Algorithmic jewellery

2011-03-19 Piling Ducks

2011-03-06 Greco de Ruijter

2011-03-05 Fractal columns

2011-02-28 Kaleidoscopic IFS

2011-02-27 Ducks and butterflies

2011-02-18 Geological artwork

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February 28th 2011

Kaleidoscopic IFS

Just after blogging about Ducks and butterflies, I found out on Fractal Forums about a 3 dimensional analogue of Jonathan McCabe's Butterfly Origami algorithm, named Kaleidoscopic IFS.

The idea is explained here, but in French. Still, it might help to look at the picture there to understand the following explanations. We start from a platonic solid. Its symmetry planes define a triangulation of the unit sphere. We pick one of these triangles and consider the three planes intersecting the sphere on its sides. They define a cone with a triangular basis. By a sucession of reflections about the three planes, we can map any point inside this cone. Let's do this. Now we'll perform a sequence of rotations and of translations. If the point gets out of the cone, we use the appropriate combination of reflexions to map it back inside. We iterate the last two steps a certain number of times. Finally, we draw as solid only those points that stayed within a certain distance of the tip of the cone during the whole process.

This yields some very cool imagery, and the patterns obtained often looks strikingly similar to their 2d analogues. Here is a bunch of links:

• A Flickr gallery by Softology
• A blog post on Syntopia
• A gallery by Jos Leys

Finally here is a very elegant video of an evolving Kaleidoscopic IFS fractal by Tom Beddard, aka subBlue.

Update: I forgott to mention the original post about Kaleidoscopic IFS fractals on Fractal Forum.